“I do not think, however, that I have even yet brought out the greatest contribution of medievalism to the formation of the scientific movement. I mean the inexpugnable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner, exemplifying general principles. Without this belief the incredible labours of scientists would be without hope. It is this instinctive conviction, vividly poised before the imagination, which is the motive power of research:—that there is a secret, a secret which can be unveiled. How has this conviction been so vividly implanted on the European mind?

When we compare this tone of thought in Europe with the attitude of other civilisations when left to themselves, there seems but one source for its origin. It must come from the medieval insistence on the rationality of God, conceived as with the personal energy of Jehovah and with the rationality of a Greek philosopher. Every detail was supervised and ordered: the search into nature could only result in the vindication of the faith in rationality. Remember that I am not talking of the explicit beliefs of a few individuals. What I mean is the impress on the European mind arising from the unquestioned faith of centuries. By this I mean the instinctive tone of thought and not a mere creed of words.

In Asia, the conceptions of God were of a being who was either too arbitrary or too impersonal for such ideas to have much effect on instinctive habits of mind. Any definite occurrence might be due to the fiat of an irrational despot, or might issue from some impersonal, inscrutable origin of things. There was not the same confidence as in the intelligible rationality of a personal being. I am not arguing that the European trust in the scrutability of nature was logically justified even by its own theology. My only point is to understand how it arose. My explanation is that the faith in the possibility of science, generated antecedently to the development of modern scientific theory, is an unconscious derivative from medieval theology.”

There are perhaps three relevant theological ideas in the medieval theology to which Whitehead refers. The first is the idea of the Universe as rational, because it is created by a rational God. Such an idea is implicit in, for example, the Timaeus of Plato, which suggests that eternally existing Platonic solids were used by the Creator as the shapes for the different kinds of atom:

The belief in rationality also prompted some good medieval work in the field of logic. However, Whitehead suggests that medieval theology also incorporated “the personal energy of Jehovah.” In particular, the “scrutability of nature” – the idea that Nature is knowable – is implicit in the medieval idea of Nature as a written book, intended to be read. Galileo famously quoted Tertullian (c. 160–225) on this point:

[Science] is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.” (Galileo, Il Saggiatore, 1623, tr. Stillman Drake)

[This passage has often been quoted, although today we would instead speak of equations and other mathematical constructs.]

Finally, there is the idea that the studying the Universe has value. Whitehead refers to belief systems which considered the world to be unintelligible. There were also other belief systems which devalued even the attempt to understand the world. The Neo-Platonists, for example, focussed their attention on mystical appreciation of the divine things “above,” which left little room for detailed study of the mundane and physical down here “below” (although it did encourage the Neo-Platonists to do mathematical work). The medievals flirted with Christian forms of Neo-Platonism, but the belief that God had created the Universe always gave the mundane and physical its own inherent value, as far as they were concerned.

The Stoics, on the other hand, believed in a cyclic Universe which was periodically destroyed, only for history to repeat itself in exact detail, like a serpent eating its own tail. There is a degree of pointlessness in such a viewpoint which perhaps discourages scientific investigation. Certainly, neither the Neo-Platonists nor the Stoics built the kind of scientific structure that Europeans began to construct in the 12th century.

Today, of course, the rationality and knowability of the Universe are largely taken for granted (except, perhaps, by Postmodernists), and more people are involved in the scientific enterprise than ever before. The spectacular success of science has made the rationality and knowability of the Universe so obvious, in fact, that it is difficult to comprehend a time, thousands of years ago, when most people thought that unpredictable chaos was all there was.

In my previous two posts, I outlined the Platonist view of mathematics, and the empiricist alternative. There are also two other alternatives:

Formalism

The truths of mathematics appear to be different in nature from the truths of physics. Formalism accepts this, but suggests that the nature of mathematics is inherently cultural. Different branches of mathematics are essentially just games with symbols and arbitrary rules – games that don’t have any particular meaning. Mathematicians simply work within the chosen rules. However, apart from the problem of the “unreasonable effectiveness of mathematics,” the idea that these rules are chosen arbitrarily runs counter to the experience of most mathematicians. When the concept of “number” was extended to include the imaginary numbers, for example, consistency with the existing rules meant that there was very little choice about how imaginary numbers behaved. In the words of mathematician Jacques Hadamard: “We speak of invention: it would be more correct to speak of discovery… Although the truth is not yet known to us, it pre-exists and inescapably imposes on us the path we must follow under penalty of going astray” (from the introduction to The Psychology of Invention in the Mathematical Field).

Many officially formalist mathematicians are Platonists at heart. Jean Dieudonné once wrote with refreshing honesty: “On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say: ‘Mathematics is just a combination of meaningless symbols,’ and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working on something real. This sensation is probably an illusion, but is very convenient.” (from “The Work of Nicholas Bourbaki,” American Mathematical Monthly, 77(2), Feb 1970, p. 134–145).

Logicism

Most mathematicians feel that the truths of mathematics are indeed in a different category from the truths of physics – that the truths of mathematics in a sense come first. Logicism is a way of rescuing this aspect of Platonism while avoiding the more mystical aspects. The basis for logicism is that logic also comes before physics – all sciences assume logical thought as a starting point. Logical truths “exist” in some sense, and logicists assume that there are no philosophical difficulties about this kind of existence. In other words, a mystical Platonic world is not needed to explain logic. Consequently, if we can provide a foundation for mathematics in terms of pure logic, we can retain all the benefits of Platonism without any of the problems.

In logicism, numbers are defined as being particular kinds of sets. Logicism began with Gottlob Frege, who published two volumes of his Die Grundgesetze der Arithmetik in 1893 and 1903. Sadly for Frege, his fellow mathematician and philosopher Bertrand Russell found a major flaw – now known as “Russell’s paradox” – in the work, just before the second volume was published. With Alfred North Whitehead, Russell was able to repair the flaw, in a three-volume work called Principia Mathematica (published in 1910, 1912, and 1913).

After 379 pages, Whitehead and Russell are well on the way to proving that 1 + 1 = 2.

The logicist programme, however, is not free of problems. First, it is extremely complex. It took Whitehead and Russell hundreds of pages of complicated logic to prove that 1 + 1 = 2. Normally, we try to explain complex things in terms of simple ones. It seems a little perverse to give such a complicated explanation of numerical facts that we understood in kindergarten. And, by including set theory as part of the basis, it isn’t really “pure logic” any more.

Second, there is more than one way of defining numbers as sets, and none of them is obviously “right.” This has led to the suggestion that sets are “what numbers could not be” (the title of an article by Paul Benacerraf in The Philosophical Review, 74, Jan 1965, pp. 47–73), and that numbers must be fundamentally different in nature from sets – if not Platonic objects satisfying certain axioms, then something else which exists in a non-contingent way.

Third, it is unclear whether logicism has actually gained anything. The starting assumption was that logic was simple and obvious, raising no philosophical problems. But if all of mathematics is hidden deep inside the structure of logic, then perhaps logic is not as simple as it first seemed. Mathematics and logic may in fact be different aspects of the same thing, but this may not make the fundamental questions about mathematical existence go away.